Gauss Sums of the Cubic Character over $GF(2^m)$: an elementary derivation

Abstract

An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field ${\mathbb{F}}_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its value is $-(-2{)}^{s/2}$).

minor changes; accepted for publication in Bulletin of the Polish Academy of Sciences, Ser. Mathematics